Sylvan is a parallel (multi-core) MTBDD library written in C. Sylvan implements parallelized operations on BDDs, MTBDDs and LDDs. Both sequential and parallel BDD-based algorithms can benefit from parallelism. Sylvan uses the work-stealing framework Lace and parallel datastructures to implement scalable multi-core operations on decision diagrams.

Sylvan is developed (© 2011-2016) by the Formal Methods and Tools group at the University of Twente as part of the MaDriD project, which is funded by NWO, and (© 2016-2017) by the Formal Methods and Verification group at the Johannes Kepler University Linz as part of the RiSE project. Sylvan is licensed with the Apache 2.0 license. The main author of the project is Tom van Dijk who can be reached via Please let us know if you use Sylvan in your projects and if you need decision diagram operations that are currently not implemented in Sylvan.

The main repository of Sylvan is A mirror is available at

Bindings for other languages than C/C++ also exist:


Sylvan has the following dependencies:

  • CMake for compiling.
  • gmp (libgmp-dev) for the GMP leaves in MTBDDs.
  • Sphinx if you want to build the documentation.

Sylvan depends on the work-stealing framework Lace for its implementation. Lace is embedded in the Sylvan distribution. Lace requires one additional library:

  • hwloc (libhwloc-dev) for pinning worker threads to processors.


It is recommended to build Sylvan in a separate build directory:

mkdir build
cd build
cmake ..
make && make test && make install

It is recommended to use ccmake to configure the build settings of Sylvan. For example, you can choose whether you want shared/static libraries, whether you want to enable statistics gathering and whether you want a Debug or a Release build.

Using Sylvan

To use Sylvan, the library and its dependency Lace must be initialized:

#include <sylvan.h>

main() {
    int n_workers = 0; // auto-detect
    lace_init(n_workers, 0);
    lace_startup(0, NULL, NULL);

    // use at most 512 MB, nodes:cache ratio 2:1, initial size 1/32 of maximum
    sylvan_set_limits(512*1024*1024, 1, 5);

    /* ... do stuff ... */


The call to lace_init initializes the Lace framework, which sets up the data structures for work-stealing. The parameter n_workers can be set to 0 for auto-detection. The function lace_startup then creates all other worker threads. The worker threads run until lace_exit is called. Lace must be started before Sylvan can be initialized.

Sylvan is initialized with a call to sylvan_init_package. Before this call, Sylvan needs to know how much memory to allocate for the nodes table and the operation cache. In this example, we use the sylvan_set_limits function to tell Sylvan that it may allocate at most 512 MB for these tables. The second parameter indicates the ratio of the nodes table and the operation cache, with each higher number doubling the size of the nodes table. Negative numbers double the size of the operation cache instead. In the example, we want the nodes table to be twice as big as the operation cache. The third parameter controls how often garbage collection doubles the table sizes before their maximum size is reached. The value 5 means that the initial tables are 32x as small as the maximum size. By default, every execution of garbage collection doubles the table sizes.

After sylvan_init_package, subpackages like mtbdd and ldd can be initialized with sylvan_init_mtbdd and sylvan_init_ldd. This allocates auxiliary datastructures.

If you enabled statistics generation (via CMake), then you can use sylvan_stats_report to report the obtained statistics to a given FILE*.

The Lace framework

Sylvan uses the Lace framework to offer ‘automatic’ parallelization of decision diagram operations. Many functions in Sylvan are Lace tasks. To call a Lace task, the variables __lace_worker and __lace_dq_head must be initialized as local variables of the current function. Use the macro LACE_ME to initialize the variables in every function that calls Sylvan functions and is not itself a Lace task.

Garbage collection and referencing nodes

Like all decision diagram implementations, Sylvan performs garbage collection. Garbage collection is triggered when trying to insert a new node and no empty space can be found in the table within a reasonable upper bound.

Garbage collection can be disabled with sylvan_gc_disable and enabled again with sylvan_gc_enable. Call sylvan_gc to manually trigger garbage collection.

To ensure that no decision diagram nodes are overwritten, you must ensure that Sylvan knows which decision diagrams you care about. Each subpackage implements mechanisms to store references to decision diagrams that must be kept. For example, the mtbdd subpackage implements mtbdd_protect and mtbdd_unprotect to store pointers to MTBDD variables.

MTBDD* allocate_var() {
    MTBDD* my_var = (MTBDD*)calloc(sizeof(MTBDD), 1);
    return my_var;

free_var(MTBDD* my_var) {

If you use mtbdd_protect you do not need to update the reference every time the value changes.

The mtbdd subpackage also implements thread-local stacks to temporarily store pointers and results of tasks:

MTBDD some_thing = ...;
MTBDD result_param1 = mtbdd_false, result_param2 = mtbdd_false;
while (some_condition) {
    mtbdd_refs_spawn(SPAWN(an_operation, some_thing, param1));
    result_param2 = CALL(an_operation, some_thing, param2);
    result_param1 = mtbdd_refs_sync(SYNC(an_operation));
    some_thing = CALL(another_operation, result1, result2);
return some_thing;

It is recommended to use the thread-local stacks for local variables, and to use the protect and unprotect functions for other variables. Every SPAWN and SYNC of a Lace task that returns an MTBDD must be decorated with mtbdd_refs_stack and mtbdd_refs_sync as in the above example.

References to decision diagrams must be added before a worker may cooperate on garbage collection. Workers can cooperate on garbage collection during SYNC and when functions create nodes or use sylvan_gc_test to test whether to assist in garbage collection. Functions for adding or removing references never perform garbage collection. Furthermore, only the mtbdd_makenode function (and other node making primitives) implicitly reference their parameters; all other functions do not reference their parameters. Nesting Sylvan functions (including sylvan_ithvar) is bad practice and should be avoided.

Warning: Sylvan is a multi-threaded library and all workers must cooperate for garbage collection. If you use locking mechanisms in your code, beware of deadlocks! You can explicitly cooperate on garbage collection with sylvan_gc_test().

Basic BDD/MTBDD functionality

In Sylvan, BDDs are special cases of MTBDDs. Several functions are specific for BDDs and they start with sylvan_, whereas generic MTBDD functions start with mtbdd_.

To create new BDDs, you can use:

  • mtbdd_true: representation of constant true.
  • mtbdd_false: representation of constant false.
  • sylvan_ithvar(var): representation of literal <var> (negated: sylvan_nithvar(var))

To follow the BDD edges and obtain the variable at the root of a BDD, you can use (only for internal nodes, not for leaves mtbdd_true and mtbdd_false):

  • mtbdd_getvar(bdd): obtain the variable of the root node of <bdd>.
  • mtbdd_gethigh(bdd): follow the high edge of <bdd>.
  • mtbdd_getlow(bdd): follow the low edge of <bdd>.

You need to manually reference BDDs that you want to keep during garbage collection (see the above explanation):

  • mtbdd_protect(bddptr): add a pointer reference to <bddptr>.
  • mtbdd_unprotect(bddptr): remove a pointer reference to <bddptr>.
  • mtbdd_refs_pushptr(bddptr): add a local pointer reference to <bddptr>.
  • mtbdd_refs_popptr(amount): remove the last <amount> local pointer references.
  • mtbdd_refs_spawn(SPAWN(...)): spawn a task that returns a BDD/MTBDD.
  • mtbdd_refs_sync(SYNC(...)): sync a task that returns a BDD/MTBDD.

It is recommended to use mtbdd_protect and mtbdd_unprotect. The C++ objects (defined in sylvan_obj.hpp) handle this automatically. For local variables, we recommend mtbdd_refs_pushptr and mtbdd_refs_popptr.

The following basic BDD operations are implemented:

  • sylvan_not(bdd): compute the negation of <bdd>.
  • sylvan_ite(a,b,c): compute ‘if <a> then <b> else <c>’.
  • sylvan_and(a, b): compute ‘<a> and <b>’.
  • sylvan_or(a, b): compute ‘<a> or <b>’.
  • sylvan_nand(a, b): compute ‘not (<a> and <b>)’.
  • sylvan_nor(a, b): compute ‘not (<a> or <b>)’.
  • sylvan_imp(a, b): compute ‘<a> then <b>’.
  • sylvan_invimp(a, b): compute ‘<b> then <a>’.
  • sylvan_xor(a, b): compute ‘<a> xor <b>’.
  • sylvan_equiv(a, b): compute ‘<a> = <b>’.
  • sylvan_diff(a, b): compute ‘<a> and not <b>’.
  • sylvan_less(a, b): compute ‘<b> and not <a>’.
  • sylvan_exists(bdd, vars): existential quantification of <bdd> with respect to variables <vars>.
  • sylvan_forall(bdd, vars): universal quantification of <bdd> with respect to variables <vars>.
  • sylvan_project(bdd, vars): the dual of sylvan_exists, projects the <bdd> to the variable domain <vars>.

A set of variables (like <vars> above) is a BDD representing the conjunction of the variables. A number of convencience functions are defined to manipulate sets of variables:

  • mtbdd_set_empty(): obtain an empty set.
  • mtbdd_set_isempty(set): compute whether the set is empty.
  • mtbdd_set_first(set): obtain the first variable of the set.
  • mtbdd_set_next(set): obtain the subset without the first variable.
  • mtbdd_set_from_array(arr, len): create a set from a given array.
  • mtbdd_set_to_array(set, arr): write the set to the given array.
  • mtbdd_set_add(set, var): compute the set plus the variable.
  • mtbdd_set_union(set1, set2): compute the union of two sets.
  • mtbdd_set_remove(set, var): compute the set minus the variable.
  • mtbdd_set_minus(set1, set2): compute the set <set1> minus the variables in <set2>.
  • mtbdd_set_count(set): compute the number of variables in the set.
  • mtbdd_set_contains(set, var): compute whether the set contains the variable.

Sylvan also implements composition and substitution/variable renaming using a “MTBDD map”. An MTBDD map is a special structure implemented with special MTBDD nodes to store a mapping from variables (uint32_t) to MTBDDs. Like sets of variables and MTBDDs, MTBDD maps must also be referenced for garbage collection. The following functions are related to MTBDD maps:

  • mtbdd_compose(dd, map): apply the map to the given decision diagram, transforming every node with a variable that is associated with some function F in the map by if <F> then <high> else <low>.
  • sylvan_compose(dd, map): same as mtbdd_compose, but assumes the decision diagram only has Boolean leaves.
  • mtbdd_map_empty(): obtain an empty map.
  • mtbdd_map_isempty(map): compute whether the map is empty.
  • mtbdd_map_key(map): obtain the key of the first pair of the map.
  • mtbdd_map_value(map): obtain the value of the first pair of the map.
  • mtbdd_map_next(map): obtain the submap without the first pair.
  • mtbdd_map_add(map, key, value): compute the map plus the given key-value pair.
  • mtbdd_map_update(map1, map2): compute the union of two maps, with priority to map2 if both maps share variables.
  • mtbdd_map_remove(map, var): compute the map minus the variable.
  • mtbdd_map_removeall(map, set): compute the map minus the given variables.
  • mtbdd_map_count(set): compute the number of pairs in the map.
  • mtbdd_map_contains(map, var): compute whether the map contains the variable.

Sylvan implements a number of counting operations:

  • mtbdd_satcount(bdd, number_of_vars): compute the number of minterms (assignments that lead to True) for a function with <number_of_vars> variables; we don’t need to know the exact variables that may be in the BDD, just how many there are.
  • sylvan_pathcount(bdd): compute the number of distinct paths to True.
  • mtbdd_nodecount(bdd): compute the number of nodes (and leaves) in the BDD.
  • mtbdd_nodecount_more(array, length): compute the number of nodes (and leaves) in the array of BDDs.

Sylvan implements various advanced operations:

  • sylvan_and_exists(bdd_a, bdd_b, vars): compute sylvan_exists(sylvan_and(bdd_a, bdd_b), vars) in one step.
  • sylvan_and_project(bdd_a, bdd_b, vars): compute sylvan_project(sylvan_and(bdd_a, bdd_b), vars) in one step.
  • sylvan_cube(vars, values): compute a cube (to leaf True) of the given variables, where the array values indicates for each variable whether to use it in negative form (value 0) or positive form (value 1) or to skip it (as dont-care, value 2).
  • sylvan_union_cube(set, vars, values): compute sylvan_or(set, sylvan_cube(vars, values)) in one step.
  • sylvan_constrain(bdd_f, bdd_c): compute the generic cofactor of F constrained by C, i.e, set F to False for all assignments not in C.
  • sylvan_restrict(bdd_f, bdd_c): compute Coudert and Madre’s restrict algorithm, which tries to minimize bdd_f according to a care set C using sibling substitution; the invariant is restrict(f, c) \and c == f \and c; the result of this algorithm is often but not always smaller than the original.
  • sylvan_pick_cube(bdd) or sylvan_sat_one_bdd(bdd): extract a single path to True from the BDD (returns the BDD of this path)
  • sylvan_pick_single_cube(bdd, vars) or sylvan_sat_single(bdd, vars) extracts a single minterm from the BDD (returns the BDD of this assignment)
  • sylvan_sat_one(bdd, vars, array): extract a single minterm from the BDD given the set of variables and write the values of the variables in order to the given array, with 0 when it is negative, 1 when it is positive, and 2 when it is dontcare.

Sylvan implements several operations for transition systems. These operations assume an interleaved variable ordering, such that source or unprimed variables have even parity (0, 2, 4...) and matching target or primed variables have odd parity (1, 3, 5...). The transition relations may be partial transition relations that only manipulate a subset of variables; hence, the operations also require the set of variables.

  • sylvan_relnext(set, relation, vars): apply the (partial) relation on the given variables to the set.
  • sylvan_relprev(relation, set, vars): apply the (partial) relation in reverse to the set; this computes predecessors but can also concatenate relations as follows: sylvan_relprev(rel1, rel2, rel1_vars).
  • sylvan_closure(relation): compute the transitive closure of the given set recursively (see Matsunaga et al, DAC 1993)

See src/sylvan_bdd.h and src/mtbdd.h for other operations on BDDs and MTBDDs.

Custom leaves

See src/sylvan_mt.h and the example in src/sylvan_gmp.h and src/sylvan_gmp.c for custom leaves in MTBDDs.

Custom decision diagram operations

Adding custom decision diagram operations is easy. Include sylvan_int.h for the internal functions. See sylvan_cache.h for how to use the operation cache.

List decision diagrams

See src/sylvan_ldd.h for operations on list decision diagrams.

File I/O

You can store and load BDDs using a number of methods, which are documented in the header files sylvan_mtbdd.h and sylvan_ldd.h.

Support for C++

See src/sylvan_obj.hpp for the C++ interface.

Table resizing

During garbage collection, it is possible to resize the nodes table and the cache. By default, Sylvan doubles the table sizes during every garbage collection until the maximum table size has been reached. There is also a less aggressive version that only resizes when at least half the table is full. This can be configured in src/sylvan_config.h. It is not possible to decrease the size of the nodes table and the cache.

Dynamic reordering

Dynamic reordening is not yet supported. For now, we suggest users find a good static variable ordering.


Simple examples can be found in the examples subdirectory. The file simple.cpp contains a toy program that uses the C++ objects to perform basic BDD manipulation. The mc.c and lddmc.c programs are more advanced examples of symbolic model checking (with example models in the models subdirectory).


Sylvan may require a larger than normal program stack. You may need to increase the program stack size on your system using ulimit -s. Segmentation faults on large computations typically indicate a program stack overflow.

I am getting the error “unable to allocate memory: ...!”

Sylvan allocates virtual memory using mmap. If you specify a combined size for the cache and node table larger than your actual available memory you may need to set vm.overcommit_memory to 1. E.g. echo 1 > /proc/sys/vm/overcommit_memory. You can make this setting permanent with echo "vm.overcommit_memory = 1" > /etc/sysctl.d/99-sylvan.conf. You can verify the setting with cat /proc/sys/vm/overcommit_memory. It should report 1.

I get errors about __lace_worker and __lace_dq_head

Many Sylvan operations are implemented as Lace tasks. To call a Lace task, the variables __lace_worker and __lace_dq_head must be initialized. Use the macro LACE_ME to do this. Only use LACE_ME locally (in a function), never globally!


T. van Dijk (2016) Sylvan: Multi-core Decision Diagrams. PhD Thesis.

T. van Dijk and J.C. van de Pol (2016) Sylvan: Multi-core Framework for Decision Diagrams. In: STTT (Special Issue), Springer.

T. van Dijk and J.C. van de Pol (2015) Sylvan: Multi-core Decision Diagrams. In: TACAS 2015, LNCS 9035. Springer.

T. van Dijk and A.W. Laarman and J.C. van de Pol (2012) Multi-Core BDD Operations for Symbolic Reachability. In: PDMC 2012, ENTCS. Elsevier.